What Great Britain could get from the South that it could not from the North was raw materials for its burgeoning industry. In fact, the Northern states were competing (although on a small scale) with industrial Britain as they themselves began to industrialize. Culturally, the South had the stronger tie to Britain since the early Southern colonists were from the English gentry or from Scotland or Ireland; the population was much more homogeneous than in the North, where colonists came from all over.
As far as Britain was concerned, the Northern colonies were all settled by religious extremists with whom the British elite had nothing in common (they are in fact, one of the many reasons religious differences forced many to flee to the New World.) By the time Henry Clinton (1738-1795) began his Southern invasion in 1779, Northern battles had already yielded only questionable British victories, and he decided to try his luck elsewhere. Unfortunately, the same guerrilla tactics that were successful in the North were successful in the South as well, even though the British had captured the major cities.
Sir Henry Clinton was a major British commander during the Revolutionary War. Under his command, the British tried to concentrate their efforts on the colonies of the South.
The main reason for this is that there were a large number of Loyalists (also known as Tories) in the South, especially in the backcountry among the people who had been "Regulators." They thought these people would help them fight and could run the colony after they recaptured it.
The other thing that made them focus on the South was the fact that it was much richer (because of all its plantations) than the North at that point. So that made it seem more important.